with
\(\,M\,\)  bending moment,
\(\,E\,\)  Euler-Young modulus, \(\,I\,\)  cross-sectional moment of inertia along the bending axis \(\,y\,\) ,
\(\,z\,\)  beam axial abscissa and \(\,\chi \,\)  elastic curvature.
The integral kernel \(\Phi _\lambda \)  can be selected among exponential, Gaussian or power-law type functions and must satisfy properties of positivity, symmetry and limit impulsivity [1]}.
A convenient choice for the averaging kernel is the special bi-exponential function
\(\Phi _\lambda (z,\bar{z})=\frac{1}{2 \lambda L}\exp \left({-\frac{|z-\bar{z}|}{\lambda L}}\right)\) 
